PROBLEM SETS 1 & 2. DUE THURSDAY 7 SEPTEMBER
Problem Set 1. Problems from Lecture 1.
Reading. Quick Calculus, Chapter 1.
Supplementary reading. Simmons, Chapter 1.
1. Given a quadratic equation of the from ax2 + bx + c = 0, we can solve for x using the
formula
√
−b ± b2 − 4ac
x=
.
2a
Using the above formula, solve the equation 4x2 − 5x − 6 = 0 for x.
2. Find f (x) if f (x + 1) = x2 − 5x + 3.
3. Graph the following functions, and give their domain and range.
(a) y = 3x − 2.
(b) y = 4x2 + 3.
4. Graph the following functions, and give their domain and range.
(a) y = 2x .
(b) y = 2x+3 .
5. Graph the following functions, and give their domain and range.
(a) y = log2 (x).
(b) y = log2 (x) + 5.
6. Graph the following functions, and give their domain and range.
(a) y = sin(x).
(b) y = sin(2x).
7. Graph the following functions, and give their domain and range.
(a) y = tan(x).
(b) y = 4 tan(x).
8. Simplify the following expressions.
(a) log10 ( x+y
).
z
x
(b) 25log25 (x+y)+log5 ( y ) .
9. Make the following computations using right triangles.
(a) For θ = π4 = 45◦ , compute sin(θ), cos(θ), and tan(θ).
(b) For θ =
π
6
= 30◦ , compute sec(θ), csc(θ), and cot(θ).
10. Simplify the following expressions (i.e. write them in terms of elementary trig functions
sin(φ), sin(θ), etc.).
(a) sin(θ + φ).
(b) cos(3θ).
1
2
PROBLEM SETS 1 & 2
Problem Set 2. Problems from Lecture 2.
Reading. Quick Calculus, pp. 50–97.
Supplementary reading. Simmons, Chapter 2, sections 2.1–2.5. Read section 2.6 if you
are interested in some applications of the derivative.
1. Compute the following limits.
(a) limθ→0 sin(5θ)
θ
(b) limθ→0 sin(3θ)
sin(4θ)
(c) limx→∞ x2x+1
2 +3x
(d) limx→∞ 2x
3x2 −2
2. Where are the following functions discontinuous?
(a) x2x+1
1
(b) x2 +x−6
3
(c) xx2+x
+1
x2 +2x
(d) x3 +2x
2 −x−2
3. Use the definition of the derivative to show that for f (x) = ax2 + bx + c, for constants
a, b, c ∈ R, the derivative is f 0 (x) = 2ax + b.
x
4. Use the definition of the derivative to find the derivative of the function f (x) = x+1
.
5. Use the definition of the derivative and the double angle formula to compute the derivative of f (θ) = cos(θ).
6. Sketch the graph of the following two functions. For each, state where it is not differentiable. p
(a) f (x) = |x|.
(b) f (x) = |x2 −
9|.2
x
if x ≤ −1,
7. Let f (x) =
What must m and b be for f (x) to be differmx + b if x > −1.
entiable at all points?
8. A penny is dropped off a ledge on the World Trade Center in New York City. The ledge
is 1024 feet above the ground. The penny falls a distance of s = 16t2 feet in t seconds.
(a) How long does the penny fall before it hits the ground?
(b) What is the average velocity at which the penny falls during the first three seconds?
9. With the same situation as in Problem [?], answer the following questions.
(a) What is the average velocity at which the penny falls during the last four seconds?
(b) What is the instantaneouse velocity of the penny when it hits the ground?
10. An oil tank is to be drained for cleaning. There are V gallons of oil left in the tank
after t minutes of draining, where V = 50(40 − t)2 .
(a) What is the average rate at which oil drains out of the tank during the first 20
minutes?
(b) What is the rate at which oil is flowing out of the tank 20 minutes after draining
begins?